Effect of Plasma Membrane Semipermeability in Making the

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Effect of plasma membrane semi-permeability in making the membrane electric double layer capacitances significant Shayandev Sinha, Harnoor Singh Sachar, and Siddhartha Das Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b02939 • Publication Date (Web): 02 Jan 2018 Downloaded from http://pubs.acs.org on January 3, 2018

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Effect of plasma membrane semi-permeability in making the membrane electric double layer capacitances significant Shayandev Sinha, Harnoor Singh Sachar, and Siddhartha Das∗ Department of Mechanical Engineering, University of Maryland, College Park, MD-20742, USA E-mail: [email protected] Abstract Electric double layers (or EDLs) formed at the membrane-electrolyte interface (MEI) and membrane-cytosol interface (MCI) of a charged lipid bilayer plasma membrane develop finitely large capacitances. However, these EDL capacitances are often much larger than the intrinsic capacitance of the membrane and all these capacitances are in series. Consequently, the effect of these EDL capacitances in dictating the overall membrane-EDL effective capacitance Cef f becomes negligible. In this paper, we challenge this conventional notion pertaining to the membrane EDL capacitances. We demonstrate that, based on the system parameters, the EDL capacitance for both the permeable and semi-permeable membranes can be small enough to influence Cef f . For the semi-permeable membranes, however, this lowering of the EDL capacitance can be much larger ensuring a reduction of Cef f by more than 20-25%. Furthermore, for the semi-permeable membranes, the reduction in Cef f is witnessed over a much larger range of system parameters. We attribute such an occurrence to the highly non-intuitive electrostatic potential distribution associated with the recently discovered phenomena

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of charge-inversion-like electrostatics and attainment of a positive zeta potential at the MCI for charged semi-permeable membranes. We anticipate that our findings will impact the quantification and the identification of a large number of biophysical phenomena that are probed by measuring the plasma membrane capacitance.

Introduction Quantifying the intrinsic capacitance of the plasma membrane of a biological cell can provide deep insights and enable monitoring of a large number of biophysical events such as egg fertilization, 1–3 synaptic vesicle fusion, 4,5 membrane retrieval or endocytosis, 5,6 activities of the receptor cells, 7 activities of secretory cells, 8,9 gating of the membrane-bound ionic channels, 10,11 etc. This capacitance, often measured under a voltage clamp condition, 4 is considered as the intrinsic capacitance of the membrane Cm ≈ 1 µF/cm2 . 11 On the other hand, the very charged nature of the plasma membrane separating the intracellular and the extracellular liquids, will enforce a development of separate electric double layers (or EDLs) at the interface of the membrane with both the liquids. 12–34 While this membrane EDL has been vital for controlling a number of physiological activities ranging from controlling cell death 35 and cellular signal transduction 36 to membrane-antibody interactions 37 and ATP synthesis by mitochondria, 38 the importance of the membrane EDL in affecting the overall effective membrane-EDL capacitance (Cef f ) has been invariably neglected. Such an approach has stemmed from the fact that the EDL capacitances (CEDL,CS or the capacitance associated with the EDL formed on the cytosol side and CEDL,ES or the capacitance associated with the EDL formed on the electrolyte side) are often identified to be much larger than Cm for biologically realistic ion concentrations and Cm , CEDL,CS , and CEDL,ES are all in series [see Fig. 1(d)] so that Cef f = [ C1m +

1 CEDL,CS

+

1 ]−1 CEDL,ES

≈ Cm . This notion of a relative

unimportance of the EDLs in deciding Cef f has been extensively proposed in the existing literature. 39–41 In the present paper, we re-visit this notion that the plasma membrane EDL will in2


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Positive Potential

CS (a)

Zero Potential

Positive Potential

ES Zero Potential

ES Zero Potential

CS

(b) Combination of σ and c∞ Zero Potential

CS ES (c)

Capacitance of EDL,CS

Capacitance of Capacitance membrane of EDL,ES

(d) Figure 1: (a) Schematic representation of electrostatic potential profile for a negativelycharged semi-permeable membrane permeable only to positive ions and demonstrating a charge-inversion (CI) like behavior in the cytosol side (CS), 33 characterized by the attainment of a positive electrostatic potential deep within the cytosol. (b) Schematic representation of the electrostatic potential profile for a negatively-charged semi-permeable membrane permeable to only positive ions and demonstrating a positive ζ potential at the MCI. Certain conditions of σ and c∞ enforce the attainment of the condition shown in (b) from the condition shown in (a). (c) Schematic representation of the electrostatic potential profile for a fully permeable membrane. (d) Schematic of the capacitances of the EDLs of the cytosol and the electrolyte sides and the intrinsic capacitance of the membrane, with all the capacitances being in series. Parts (a) and (b) of this figure have been reprinted from Sinha et al. 34 with the permission of AIP Publishing.

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evitably be unimportant in deciding this Cef f . Rather, through a simple Poisson-Boltzmann (PB) model, we establish that in the presence of certain values of the system parameters such as the salt concentration (c∞ ) and the membrane surface charge density (σ), CEDL for both the permeable and semi-permeable membranes can become significantly small to affect Cef f . We consider a particular type of a semi-permeable membrane – a negatively-charged semipermeable membrane permeating only positive ions from the electrolyte side to the cytosol side. We demonstrate that the extent of lowering of CEDL (and consequently, Cef f ) can be significantly more enhanced for the semi-permeable membrane as compared to the fully membrane membrane. Moreover, the range of the system parameters over which this lowering of Cef f is encountered is significantly larger for the semi-permeable membrane. These are the central results of this paper. Our analysis first probes the electric double layer electrostatic potential (ψ) for both the permeable and semi-permeable membranes [see Figs. 1(a-c) for the schematics]. In two recent papers, 33,34 we have computed this ψ for both fully permeable and semi-permeable plasma membranes – a negatively-charged semi-permeable plasma membrane permeating only positive ions demonstrate extremely weird charge inversion (CI) like electrostatics 32,33 and develops a positive ζ potential at the membrane-cytosol-interface (MCI). We use this ψ information to compute CEDL,CS and CEDL,ES and demonstrate that the presence of CI-like electrostatics in the cytosol side or the attainment of a positive ζ potential at the MCI makes CEDL,CS small enough so as to ensure that Cef f gets massively affected by the EDL capacitance. Complete ignorance of such highly non-trivial semi-permeable membrane EDL electrostatics, revealed only very recently theoretically, 32–34 has forbidden identification of such situations where CEDL can decisively influence Cef f . It is worthwhile to note that under any measurement technique, one can measure Cef f and not Cm . This is due to the fact that it is not possible to decouple the effect of the nanometer-thick EDLs bound to the membrane on its either side. For cases where CEDL affects Cef f , Cm < Cef f – hence equating Cef f (measured) to Cm would imply an underprediction of Cm . This measured Cm (or Cef f ) value is used to predict a variety of membrane parameters (e.g., membrane surface area) and

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physiological events associated with the membrane (see above). All these parameters and events would therefore be erroneously predicted on account of the underprediction of Cm . Therefore, we anticipate that our present paper, which will also be useful in providing the foundation for understanding the EDL electrostatics for membranes with curvatures (e.g., nanoparticle-supported lipid bilayer 42–44 ), would motivate the biophysicists and biologists to more carefully infer about the different biophysical phenomena on the basis of the measurement of membrane capacitance under those conditions of membrane semi-permeability, membrane surface charge density and salt concentration for which CEDL affects Cef f .

Theory General EDL theory for the capacitance of the plasma-membraneEDL system In this subsection, we shall propose a general theory for the electrostatics and the capacitances of the plasma-membrane-EDL system employing the standard Gouy-Chapman Poisson-Boltzmann model. The equations for the EDL electrostatic potential (ψ) and the relevant boundary conditions have already been discussed in our previous papers. 33,34 We repeat them here for the sake of continuity. We consider a plasma membrane, as shown in Fig. 1(a-c). The membrane consists of a lipid bilayer with the bilayers having no volume charge. However, the hydrophilic heads of the bilayers are charged with a charge density of σ (in C/m2 ). We consider σ < 0. Therefore, both the MCI and the membrane-electrolyte interface (MEI) have a surface charge density of σ. Both the cytosol and the electrolyte contain ions and form individual EDLs at their interfaces with the membrane [see Fig. 1(a-c)] – the nature of this EDL is governed by the membrane permeability. In this subsection, we shall provide a general theory that is valid for both permeable and semi-permeable membranes, or in other words valid regardless of the cytosol and the electrolyte EDL compositions. We consider that the EDL electrostatic potential ψ is governed by the Poisson equation as 5


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described below: d2 ψ ρe,c = − dy 2 0 c 2 dψ =0 dy 2 d2 ψ ρe,e = − dy 2 0 e

f or − (dm + dc ) ≤ y ≤ −dm , f or − dm < y < dm , f or dm ≤ y ≤ (dm + de ).

(1)

In the above equations, ρe,c and ρe,e are the charge densities of the cytosol and the electrolyte EDLs respectively, c and e are the relative permittivites of the cytosol and the electrolyte, 0 is the permittivity of free space, 2dm is the membrane thickness, and 2de and 2dc are the thicknesses of the electrolyte and the cytosol. Eq.(1) needs to be solved in presence of the following boundary conditions: dψ dψ σ m − c =− , dy y=−d+m dy y=−d−m 0 dψ dψ σ e − m =− , dy y=d+m dy y=d−m 0 dψ = 0, (ψ)y=−d+m = (ψ)y=−d−m , dy y=−(dm +dc ) dψ = 0. (ψ)y=d+m = (ψ)y=d−m , dy y=(dm +de )

(2)

In the above equation, m is the relative permittivity of the membrane. Under these conditions, the charge density within the cytosol EDL can be expressed as [using

6


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eqs.(1,2)]: Z

−dm

ρe,c dy =

σEDL,CS = −(dm +dc )

Z

−dm

−0 c −(dm +dc )

d2 ψ dy 2

dy =

"

# dψ dψ −0 c − = dy y=−dm dy y=−(dm +dc ) dψ dψ = −σ − 0 m . −0 c dy y=−dm dy y=−dm

(3)

Similarly, the net charge density within the electrolyte EDL can be expresses as [using eqs.(1,2)]: Z

dm +de

ρe,e dy = Z dψ −0 e dy = dy 2 dm # " dψ dψ = −0 e − dy y=dm +de dy y=dm dψ dψ −0 e = −σ + 0 m . dy y=dm dy y=dm σEDL,ES =

dm dm +de 2

(4)

Finally, the cytosol EDL capacitance can be expressed as [using eq.(3)]

CEDL,CS = |

σEDL,CS |= ψy=−dm − ψy=−(dm +dc ) σ | |, ψy=−dm − ψy=−(dm +dc )

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(5)

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while the electrolyte EDL capacitance can be expressed as [using eq.(4)]

CEDL,ES = |

σEDL,ES |= ψy=dm − ψy=dm +de σ |. | ψy=dm − ψy=dm +de

(6)

From eqs.(3-6) it is apparent that we would need the ψ profile to compute CEDL,CS and CEDL,ES . ψ profiles for both the cases of fully permeable and semi-permeable membranes are discussed later. Given that the three capacitances, namely the cytosol EDL capacitance (CEDL,CS ), intrinsic capacitance of the membrane (Cm ) and the electrolyte EDL capacitance (CEDL,ES ) are all in series, we may obtain the effective capacitance Cef f as:

Cef f =

1 1 CEDL,CS

+

1 Cm

+

.

1

(7)

CEDL,ES

Case 1: Case of fully-permeable membrane For this case, we consider a salt AB of bulk concentration c∞ (or bulk number density of n∞ = 103 NA c∞ , where NA is the Avogadro number) where the membrane is permeable to both monovalent cation A+ and monovalent anion B − . We consider the membrane to be negatively charged (i.e., σ < 0), and consequently the EDLs formed both in the cytosol and the electrolyte side consists of A+ ions as counterions and B − ions as coions. As a result, one can write, employing Boltzmann distribution for the cations and the anions: ρe,c = ρe,e = e (nA+ − nB − ) = −2n∞ e sinh

eψ . kB T

(8)

We obtain the EDL electrostatic potential profile ψ by using eq.(8) in eq.(1) and then solving ψ numerically in presence of the boundary conditions expressed in eq.(2). Once ψ is obtained, we use eqs.(5,6,7) to obtain the final value of the effective capacitance of the plasma-membrane-EDL system for a fully permeable plasma membrane.

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Case 2: Case of a semi-permeable membrane Here we consider the case of a negatively charged plasma membrane permeable to only positive ions. We consider the salt AD to be present in the electrolyte side. The membrane is semi-permeable to the salt AD, i.e., it permeates the cation A+ , but does not permeate the anion D− . Therefore, there is no D− ions in the cytosol side. The bulk concentration of AD is c∞ (or equivalently a bulk number density of n∞ = 103 NA c∞ ). Under these conditions, one may write (using Boltzmann distribution): ρe,e = e (nA+ − nD− ) = −2en∞ sinh

ρe,c = enA+

eψ , kB T

eψ = en∞ exp − . kB T

(9)

(10)

For this case, ψ can be obtained by using eqs.(9,10) in eq.(1) and then solving the resulting equation numerically in presence of the boundary conditions expressed in eq.(2). Once ψ has been obtained, Cef f can be computed using eqs.(5,6,7).

Results and Discussions Capacitance for fully permeable plasma membrane (Case 1) Fig. 2 shows the electrostatic potential profile ψ and the corresponding capacitances CEDL,CS , CEDL,ES (please note that for a fully semi-permeable membrane, CEDL,CS = CEDL,ES ) and Cef f for a fully permeable membrane as a function of c∞ and σ. This potential profile, shown in Figs. 2(a-c) has already been partly discussed in our previous paper; 33 we repeat the discussion here for the sake of continuity. The membrane being fully permeable it supports a standard EDL (composed of both coions and counterions) on both the MCI and the MEI. Accordingly, we get a perfectly symmetric ψ distribution. Moreover, a de9


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crease in the salt concentration increases the EDL thickness (λ) on both the electrolyte and the cytosol sides and also leads to a larger magnitude of (identical) ψ at the MEI and the MCI. Furthermore, the EDLs at both the MCI and the MEI being standard EDLs formed of both coions and counterions and the fact that dc λ and de λ, the condition of (dψ/dy)−dm −dc = (dψ/dy)dm +de = 0 would ensure (ψ)−dm −dc = (ψ)dm +de → 0. This implies that the electrostatic potential is almost zero both deep within the electrolyte as well as deep within the cytosol. Consequently, a larger magnitude of ψ at the MEI (or MCI) due to weaker c∞ (or a larger λ) for a given σ would imply a larger value of∆ψ (or the magnitude of the total potential drop) within the electrolyte (or the cytosol) for a smaller c∞ . Accordingly, a smaller c∞ would lead to a smaller value of CEDL,CS or CEDL,ES [see Fig. 2(d)] and hence a smaller value (< 1) of Cef f /Cm [see Fig. 2(e)]. On the other hand, a larger σ would lead to a larger ψ at the MEI and the MCI [see Fig. 2(a-c)] and therefore would lead to a larger potential drop across the electrolyte and the cytosol. However, the increase in σ counters this effect of the increase in the potential drop in dictating the overall EDL capacitance value. For a very large σ, the effect of the enhancement in σ overwhelms the effect of the increase in the potential drop. On the other hand, for a smaller σ, these two effects are pretty similar. Therefore, one witnesses a much larger increase in CEDL as σ increases from 0.1 C/m2 to 1 C/m2 as compared to what happens when σ increases from 0.01 C/m2 to 0.01 C/m2 . This variation in CEDL eventually ensures a significantly weak decrease of Cef f for large σ, but substantially noticeable decrease in Cef f for weaker σ. However, even for weaker σ, the EDL-mediated lowering of Cef f for a fully permeable membrane is witnessed only at a significantly weak value of (≤ 10−2 M ) of the salt concentration.

Capacitance for a semi-permeable plasma membrane (Case 2) Fig. 3 shows the electrostatic potential profile (ψ) and the capacitances CEDL,CS , CEDL,ES , and Cef f for the negatively charged semi-permeable plasma membrane permeating only positive ions (from the electrolyte to the cytosol side) for different combinations of σ and 10


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0.5

0

0

-2

2

-0.5 ¯ ψ

¯ ψ

σ=-1 e/nm

σ=-0.1 e/nm

c ∞=0.01 M

-8

-6

-4

-2

0

2

4

6

∞

c =0.1 M ∞

c =1 M

-2

c ∞=1 M

-6

c =0.01 M

-1.5

c ∞=0.1 M

8

-2.5 -8

∞

-6

-4

-2

0

y¯

4

6

8

(b)

300

(c) 1

2

0.95

2

σ=-0.01 e/nm

2

Cef f /Cm

σ=-1 e/nm

σ=-0.1 e/nm

200

2

y¯

(a) 250

2

-1

-4

CEDL,C S /Cm & CEDL,ES /Cm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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150

0.9 2

σ=-1 e/nm 2 σ=-0.1 e/nm

0.85

100

σ=-0.01 e/nm 2

0.8

50 0 10 -3

10 -2

10 -1

10 0

0.75 -3 10

10

-2

10

-1

10

0

c∞ (M)

c∞ (M)

(d)

(e)

Figure 2: Electrostatic potential profiles and capacitances of a negatively charged fully permeable plasma membrane.Electrostatic potential profiles as a function of the salt concentration (c∞ ) are provided for (a) σ = −1e/nm2 , (b) σ = −0.1e/nm2 , and (c) σ = −0.01e/nm2 . (d) Variation of the capacitances (made dimensionless with the intrinsic membrane capacitance Cm ) associated with the EDLs in the CS (CEDL,CS ) and the ES (CEDL,ES ) with c∞ for different values of σ. The membrane being fully permeable, the EDL on the ES is identical to the EDL on the CS, making CEDL,ES = CEDL,CS . (e) Variation of the membrane-EDL effective capacitance Cef f , made dimensionless with Cm , with c∞ for different values of σ. For these plots, we consider ψ¯ = eψ/(kB T ), y¯ = y/dm , Cm = 1 µF/cm2 , dm = 4 nm, dc = de = 1 µm, 0 = 8.8 × 10−12 F/m, e = c = 79.8, m = 3.9, e = 1.6 × 10−19 C, kB T = 4.11 × 10−21 J.

c∞ . Like the case of the permeable membrane, for this case as well, the ψ profile [shown in Fig. 3(a-c)] has already been discussed in our previous papers 33,34 and we repeat it here for the sake of continuity and better explanation of the variation of the capacitance. In comparison to the case of a fully permeable membrane, here we witness distinctly larger

11


8

5

6

4

2

2

ψ¯

3

0

5

σ=-0.1e/nm

c =0.01M ∞

3

1

c ∞=1M

-6 -5

0

5

c ∞=0.01M

-3

∞

c ∞=1M

-10

-5

0

5

c ∞=1M

0

c ∞=0.1M

-2 10

c ∞=0.01M c =0.1M

-1

10

-10

-5

0

y¯

y¯

y¯

(a)

(b)

(c)

400

80

σ= -1e/nm

CEDL,ES /Cm

60

σ= -0.1e/nm2 σ= -0.01e/nm2

40 20

300

10

1 σ= -1e/nm

2

5

2

σ= -0.1e/nm2 σ= -0.01e/nm2

Cef f /Cm

-10

2 1

-1

c ∞=0.1M

σ=-0.01e/nm 2

4

2

0

-4

-8

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ψ¯

σ=-1e/nm 2

4

-2

CEDL,CS /Cm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ψ¯

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200

0.9

0.8

σ= -1e/nm2

100

σ= -0.1e/nm2 σ= -0.01e/nm2

0 10 -3

10 -2

10 -1

10 0

0 10 -3

10 -2

10 -1

10 0

0.7 10 -3

10 -2

10 -1

c∞(M)

c∞(M)

c∞(M)

(d)

(e)

(f)

10 0

Figure 3: Electrostatic potential profiles and capacitances of a negatively charged semipermeable plasma membrane permeating only positive ions from the electrolyte side (ES) to the cytosol side (CS). Electrostatic potential profiles as a function of the salt concentration (c∞ ) are provided for (a) σ = −1e/nm2 , (b) σ = −0.1e/nm2 , and (c) σ = −0.01e/nm2 . (d) The capacitance of the EDL on the CS is represented with respect to the concentration of the electrolyte in comparison to the capacitance of the membrane (taken as 1µF/cm2 ). (d) Variation of the capacitance associated with the EDL on the cytosol side CEDL,CS , made dimensionless with Cm with the salt concentration (c∞ ) for different σ. (e) Variation of the capacitance associated with with the EDL on the electrolyte side CEDL,ES , made dimensionless with Cm , with the salt concentration (c∞ ) for different σ. (f) Variation of the membrane-EDL effective capacitance Cef f , made dimensionless with Cm , with the salt concentration (c∞ ) for different σ. For these plots, we consider ψ¯ = eψ/(kB T ) and y¯ = y/dm . All other parameters are identical to that used in Fig. 2. Parts (a-c) in this figure have been reprinted from Sinha et al. 34 with the permission of AIP Publishing.

EDL-mediated decrease of Cef f /Cm . This decrease can be as large as 25% (or even more) for small (but experimentally supported 45–48 ) values of σ and c∞ . Secondly, unlike the fully permeable membrane, for the semi-permeable membrane a finite decrease in Cef f is witnessed 12


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even for a significantly large range of electrolyte salt concentration. These are the central results of the paper and to the best of our knowledge show for the first time such a significant influence of the EDL capacitance in the overall EDL-plasma-membrane effective capacitance. The semi-permeable nature of the membrane, allowing only positive ions to permeate from the ES to the CS, leads to the formation of a counterion-only EDL at the MCI 33,34 and is responsible for several highly non-intuitive phenomena such as CI-like electrostatics 32,33 and attainment of a positive ζ potential at the MCI. 34 This CI-like electrostatics refers to a situation where the EDL electrostatic potential deep within the cytosol (i.e., at a location where the net charge content is zero, i.e., dψ/dy = 0) becomes positive despite the MCI being negative charged and having a negative ζ potential [see Fig. 3(a)]. An even more non-intuitive occurrence is the development of a positive zeta potential at the MCI itself, which we attribute to the development of a steep constant electrostatic potential gradient within the membrane [see Fig 3(b,c)]. Occurrence of such a positive ζ potential is witnessed typically for weak σ and large salt concentration. The electrolyte side, on the other hand, bears a standard EDL (consisting of both coions and counterions) and does not demonstrate such non-intuitive effects. Therefore, CEDL,ES remains significantly high (as for the case of the fully permeable membrane) [see Fig. 3(e)], while CEDL,CS , on account of the effect like CI-like electrostatics and positive ζ potential at the MCI, encounters a distinct lowering [see Fig. 3(e)] that eventually ensures a lowering of Cef f /Cm [see Fig. 3(f)]. For relatively small σ and small c∞ , Cef f for the semi-permeable membrane is distinctly smaller than Cef f for the fully permeable membrane. For such a combination of σ and c∞ , the cytosol side demonstrates a distinct CI-Like electrostatics [see Fig. 3(a)], characterized by a negative ζ potential at the MCI and a large positive electrostatic potential deep within the cytosol. Consequently, one encounters a large |∆ψ| across the EDL supported at the MCI. This large |∆ψ|, coupled with a weak σ, eventually leads to a small CEDL,CS [see Fig. 3(d)] ensuring a significant lowering of Cef f [see Fig. 3(f)]. Increase in salt concentration for such weak values of σ eventually leads to an attainment of a positive ζ potential at the

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MCI [see Fig. 3(b,c)]. 34 The potential deep within the cytosol is also positive. Hence the corresponding |∆ψ| is much smaller. Accordingly, CEDL,CS is larger [see Fig. 3(d)] effecting a much smaller decrease in Cef f . This explains the increase in Cef f with an increase in c∞ . However, despite such a decrease, the extent of the reduction of Cef f for the semi-permeable membrane remains significantly larger at larger salt concentration as compared to the case of a fully permeable membrane. Decrease in σ decreases the jump in |∆ψ| in both the CS and ES; however, given that |∆ψ| for the CS is dictated by the attainment of either CILike electrostatics (for lesser c∞ ) or the attainment of a positive ζ potential at the MCI (for a larger c∞ ), this decrease for the CS is not as strong as the case of a fully permeable membrane. Consequently, the effect of the reduction of σ dictates CEDL,CS [see Fig. 3(d)], ensuring a reduction in Cef f with a reduction in σ [see Fig. 3(f)].

σ − c∞ phase space governing the reduced Cef f Finally in Fig. 4, we provide the σ-c∞ phase-space that shows the region where Cef f /Cm < 0.8, i.e., there is at least 20% reduction in the effective capacitance on account of the finite contribution of the membrane EDL capacitances. We consider the cases of both the fully permeable and semi-permeable membranes. This phase-space gives us a clear idea about the operating range of concentration and charge density of the membrane where the effective capacitance will get influenced by the properties of the membrane and the EDL formed around the membrane. Examples of membranes operating at these regimes of concentration can be found in literature 47,48 and from Fig. 4 it is intuitive the manner in which the EDL effects are significantly more important in affecting Cef f for the semi-permeable membrane.

Gouy-Chapman-Stern model for the membrane EDL electrostatics The biologically relevant conditions, as considered here, imply significantly large values of σ and c∞ , which necessitate the use of the Gouy-Chapman-Stern (GCS) model rather the simplistic Gouy-Chapman model, as has been considered here. In the GCS model, one accounts 14


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-10 -3

σ(e/nm2 )

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Ceff/Cm 0, attributable to the semi-permeability-driven development of counterion-only EDL at the CS, has not been identified previously. This has forbidden the identification of this situation where the EDL capacitance may become significantly small giving rise to a most noteworthy situation where the membrane EDL capacitance may significantly influence the overall plasma-membrane-EDL-capacitance. Supporting Information Available: In the Supporting Material, we develop a detailed Gouy-Chapman-Stern model to describe the membrane EDL electrostatics and the resulting membrane-EDL effective capacitance. This material is available free of charge via the Internet at http://pubs.acs.org.

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Effect of Plasma Membrane Semipermeability in Making the

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